Nuclear Force from Lattice QCD

The nucleon-nucleon ($NN$) potential is studied by lattice QCD simulations in the quenched approximation, using the plaquette gauge action and the Wilson quark action on a ${32}^4$ [$\simeq (4.4\mathrm{fm}{)}^4$] lattice. A $NN$ potential $V_{NN}(r)$ is defined from the equal-time Bethe-Salpeter amplitude with a local interpolating operator for the nucleon. By studying the $NN$ interaction in the ${}^{1}S_0$ and ${}^{3}S_1$ channels, we show that the central part of $V_{NN}(r)$ has a strong repulsive core of a few hundred MeV at short distances ($r\lesssim 0.5\mathrm{fm}$) surrounded by an attractive well at medium and long distances. These features are consistent with the known phenomenological features of the nuclear force.

Figure 1

Three examples of the modern $NN$ potential in the ${}^{1}S_0$ (spin singlet and $s$-wave) channel: CD-Bonn [17], Reid93 [18], and AV18 [19] from the top at $r=0.8\mathrm{fm}$.

Figure 2

The lattice QCD result of the radial dependence of the $NN$ wave function at $t-t_0=6$ in the ${}^{1}S_0$ and ${}^{3}S_1$ channels. Inset shows the two-dimensional view in the $x-y$ plane.

Figure 3

The lattice QCD result of the central (effective central) part of the $NN$ potential $V_C(r)$ [$V_C^{\mathrm{eff}}(r)$] in the ${}^{1}S_0$ (${}^{3}S_1$) channel for $m_{\pi }/m_{\rho }=0.595$. The inset shows its enlargement. The solid lines correspond to the one-pion exchange potential (OPEP) given in Eq. (5).

Figure 4

$t-t_0$ dependence of $V_C(r)$ in the ${}^{1}S_0$ channel for several different values of the distance $r$.